The following sequence is given below:
`1,3,3,3,5,5,5,5,5,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,……`
The 111th term of this sequence will be :

Options

A. a. 111
B. b. 19
C. c. 21
D. d. none of these

Answer

To find the 111th term of the given sequence `1, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, …`, we can analyze the pattern in the sequence.

### Step-by-Step Solution:

1. **Identify the Pattern:**
The sequence consists of odd numbers where each odd number appears a number of times equal to its value. For example:
- The number `1` appears `1` time.
- The number `3` appears `3` times.
- The number `5` appears `5` times.
- The number `7` appears `7` times.
- The number `9` appears `9` times.
- The number `11` will appear `11` times, and so on.

2. **Calculate the Total Number of Terms:**
We need to calculate how many terms are there in the sequence up to a certain odd number. We can sum the occurrences of each odd number until we reach or exceed 111 terms.

- For `1`: 1 term (Total = 1)
- For `3`: 3 terms (Total = 1 + 3 = 4)
- For `5`: 5 terms (Total = 4 + 5 = 9)
- For `7`: 7 terms (Total = 9 + 7 = 16)
- For `9`: 9 terms (Total = 16 + 9 = 25)
- For `11`: 11 terms (Total = 25 + 11 = 36)
- For `13`: 13 terms (Total = 36 + 13 = 49)
- For `15`: 15 terms (Total = 49 + 15 = 64)
- For `17`: 17 terms (Total = 64 + 17 = 81)
- For `19`: 19 terms (Total = 81 + 19 = 100)
- For `21`: 21 terms (Total = 100 + 21 = 121)

3. **Determine the Range for the 111th Term:**
From the calculations, we see that:
- The total number of terms up to `19` is `100`.
- The total number of terms up to `21` is `121`.

Therefore, the 111th term must be within the occurrences of `21`, since it falls between `100` and `121`.

4. **Find the Exact Position of the 111th Term:**
Since the 111th term is the `11th` term after the `100th` term, we can conclude that:
- The 101st term is `21`
- The 102nd term is `21`
- The 103rd term is `21`
- The 104th term is `21`
- The 105th term is `21`
- The 106th term is `21`
- The 107th term is `21`
- The 108th term is `21`
- The 109th term is `21`
- The 110th term is `21`
- The 111th term is `21`

Thus, the **111th term of the sequence is `21`**.